Vectors and Projectiles - Mission VP4 Detailed Help

A city jogger runs 6 blocks due east and then 11 blocks due north. The direction of the jogger's displacement is ____ degrees. (Use the counterclockwise from East convention.) 
(Note: Numbers are randomized numbers and likely different from the numbers listed here.)

The head-to-tail method of vector addition should be used to create a rough sketch of this physical situation. In this method, the first vector (6 blocks, east) is sketched (not to scale) in its indicated direction. The second vector (11 blocks, north) is then sketched (not to scale) starting at the head (arrowhead) of the first vector. The resultant vector is then drawn from the tail of the first vector to the head of the last vector. A trigonometric function can then be used to calculate the angle which the axis makes with one of the nearest axes (see Math Magic section).

The trigonometric functions sine, cosine and tangent can be used to express the relationship between the angle of a right triangle and the lengths of the adjacent side, opposite side and hypotenuse. The meaning of the three functions are:
sine Θ = (length of opposite side / length of hypotenuse)
cosine Θ = (length of adjacent side / length of hypotenuse)
tangent Θ = (length of opposite side / length of adjacent side)

An effective strategy for all questions in this sublevel will center around a rough sketch of the addition of two vectors (See Think About It). Consider the following steps:
  • Sketch the first vector in the appropriate direction. Place an arrowhead at the end of the vector and label the vector's magnitude on the sketch.
  • Starting at the arrowhead of the first vector, draw the second vector in the appropriate direction and to the approximate length. Put an arrowhead at the end of the vector and label the vector's magnitude on the sketch.
  • Draw the resultant vector from the tail of the first to the arrowhead of the second vector. Label the vector as R (for resultant) and put an arrowhead at the end of the resultant vector.
  • Label the angle between the resultant vector and the horizontal leg of the triangle as angle Theta.
  • Since the two vectors being added and the resultant form a right triangle, SOH CAH TOA can be used to calculate the angle Theta (see Math Magic section). The tangent function can be used to relate the angle to the length of the horizontal and vertical legs of this right triangle.
  • If the angle Thetais the angle between the resultant R and the east direction, then Thetais the answer to the question. If it is not the angle between east and the resultant, then you will have to use  and your sketch to determine the direction of the vector (see the Dictionary section).

Counterclockwise from East Convention for Vector Direction
The direction of a vector is often expressed using the counterclockwise (CCW) convention. According to this convention, the direction of a vector is the number of degrees of rotation which the vector makes counterclockwise from East.


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